Zhang

The Zhang potential is a coarse-grained, one-particle-thick model for biological and biomimetic fluid membranes developed by Yuan et al. [Yuan2010] . It features an anisotropic pair potential that allows particles to self-assemble into fluid membranes with biologically relevant properties.

The potential consists of two parts:

1. A distance-dependent function \(U(r)\)

2. An orientation-dependent function \(\Omega(r̂_{ij}, n_{i}, n_{j})\)

The total potential is given by:

\[\begin{split}U(\mathbf{r}_{ij}, \mathbf{n}_i, \mathbf{n}_j) = \begin{cases} U_R(r) + \varepsilon[1 - \Omega(\hat{\mathbf{r}}_{ij}, \mathbf{n}_i, \mathbf{n}_j)], & r \leq r_{\text{min}} \\ U_A(r)\Omega(\hat{\mathbf{r}}_{ij}, \mathbf{n}_i, \mathbf{n}_j), & r_{\text{min}} < r \leq r_c \end{cases}\end{split}\]

The distance-dependent functions:

\[U_R(r) = \varepsilon \left[\left(\frac{r_{\text{min}}}{r}\right)^4 - 2\left(\frac{r_{\text{min}}}{r}\right)^2\right]\]

\[U_A(r) = -\varepsilon \cos^2\left[\frac{\pi}{2}\frac{r - r_{\text{min}}}{r_c - r_{\text{min}}}\right]\]

The orientation-dependent function:

\[\Omega = 1 + \mu[(\mathbf{n}_i \cdot \hat{\mathbf{r}}_{ij})(\mathbf{n}_j \cdot \hat{\mathbf{r}}_{ij}) + \sin\theta_0(\mathbf{n}_j - \mathbf{n}_i) \cdot \hat{\mathbf{r}}_{ij} - \sin^2\theta_0]\]

where \(r_{ij}\) is the distance between particles, \(n_i\) and \(n_j\) are unit vectors representing particle orientations, \(r_{\text{min}}\) is the distance of minimum energy, and \(r_c\) is the cutoff distance.

---

• type: NonBonded, Zhang

• parameters:

  • alpha: real: Controls the slope of the attractive potential branch

  • beta: real: Related to membrane bending rigidity

  • gamma: real: Related to membrane spontaneous curvature

  • r_min: real: Distance of minimum energy [distance]

  • r_c: real: Cutoff distance [distance]

  • axis: real3: Reference axis for particle orientations (default: {0, 0, 1})

• data:

  • name_i: string: Type of particle i

  • name_j: string: Type of particle j

  • epsilon: real: Energy scale [energy]

Example:

"zhang":{
  "type":["NonBonded","Zhang"],
  "parameters":{
    "alpha": 4.0,
    "beta": 3.0,
    "gamma": 0.0,
    "r_min": 1.1225,
    "r_c": 2.6,
    "axis":[0, 0, 1]
  },
  "labels":["name_i", "name_j", "epsilon"],
  "data":[
    ["A", "A", 1.0],
    ["A", "B", 1.0],
    ["B", "B", 1.0]
  ]
}

Note

This potential requires orientation information for each particle. Ensure that the orientation (director) is properly set for each particle before using this potential.

[Yuan2010]

Yuan, H., Huang, C., Li, J., Lykotrafitis, G., & Zhang, S. (2010). One-particle-thick, solvent-free, coarse-grained model for biological and biomimetic fluid membranes. Physical Review E, 82(1), 011905.